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Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

19229 questions
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Prove that a complex equation has a solution of module 1

Prove that the equation $$z^n + z + 1=0 \ z \in \mathbb{C}, n \in \mathbb{N} \tag1$$ has a solution $z$ with $|z|=1$ iff $n=3k +2, k \in \mathbb{N} $. One implication is simple: if there is $z \in \mathbb{C}, |z|=1$ solution for (1) then…
user261263
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3 answers

How do you build the complex numbers from foundations?

Id like to know how the complex numbers are derived from axioms. Most textbooks talk of them as starting from the complex plane. Is there a way to derive the complex properties from say the properties of fields or is there some peano axiom like…
jim
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How can I express $ i^{2i}$ in the form $x + iy$?

I'm not sure how to begin since this is not in the form $re^{i \theta}$.
Edi Madi
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Multiplying two radicals with negatives, simple algebra?

Evaluate $$ \sqrt{-9}\sqrt{-4} $$ Now, I am told that $\sqrt{a}\sqrt{b}=\sqrt{ab}$, so I should be able to simply write $$ \sqrt{-9}\sqrt{-4} = \sqrt{(-9)(-4)}=\sqrt{36} = 6 $$ However, I am also told that I can simplify things like $\sqrt{-9}$ to…
Carser
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A proof about complex number

If $a, b, c\in \mathbb{C}$, and if $\left \| a \right \|=\left \| b \right \|=\left \| c \right \|=1$, prove $(a+b)(b+c)(c+a)/(abc)\in \mathbb{R}$. I have thought this Q for a long time, but I can only get something long and troublesome but not the…
JSCB
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How many elements are in the set $ \{ \left( \frac{2+i}{2-i} \right) ^n : n \in \mathbb N\}$

How many elements are in the set $$ \left\{ \left( \frac{2+i}{2-i} \right) ^n : n \in \mathbb N\right\}$$ My attempt: $\left( \frac{2+i}{2-i} \right) ^n = \left( \frac{3}{5} + \frac{4}{5}i \right)^n=e^{\arctan(4/3)ni}$ So if for $n_1 < n_2$ it is…
zesy
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Solving $\lvert z \rvert z^2 = \sqrt{2}(1-i)\overline{z}$

I haven't been able to solve the complex equation $$\lvert z \rvert z^2 = \sqrt{2}(1-i)\overline{z},$$despite trying writing $z$ in different forms and using $z\overline{z}=\lvert z\rvert^2$. I'm missing something again, what is it?
Richard
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Derive identities for $\cos(4x)$ and $\sin(4x)$ using following fact

So I need to use the fact that: $$\cos(4x) + i\sin(4x) = \left(\cos(x) + i\sin(x)\right)^4$$ to derive identities for $\cos(4x)$ and $\sin(4x)$ in terms of $\cos(x)$ and $\sin(x)$. I'm not sure how to go about this, could I please get some help.
Drew
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Complex numbers- elementary question

How can I calculate the argument of the complex number $z= (\frac{1}{2}+ \frac{i\sqrt{3}}{2}) \cdot (1+i)$? I always get $\tan^{-1}(-2-\sqrt{3})$, but the book answer is $7 \pi/12$.
user202072
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Fourth roots Complex analysis

I am trying to find the fourth roots of $8\sqrt2(1+i)$. So then, I was deciding to convert $1+i$ to an $re^{i\theta}$, where $r = \sqrt2$ and $\theta = 45^\circ$ or $\pi/4$. then; $z = \sqrt2e^{i\pi/4}$, but then we need the fourth roots so then,…
mary
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Compute all possible values of $\cot{\theta} - \frac{6}{z}$. Note that $ 0 < \theta < \frac{\pi}{2}$.

Let $\theta = \arg{z}$ and suppose $z$ satisfies $|z - 3i| = 3$. Compute all possible values of $\displaystyle \cot{\theta} - \frac{6}{z}$. Note that $\displaystyle 0 < \theta < \frac{\pi}{2}$. $\bf{My\; Try::}$ Let $z-3i=3e^{i\theta}\Rightarrow…
juantheron
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Absolute value of complex number

This question might be very simple, but I can't visualize how to get the absolute value of this complex number ($j$ is the imaginary unit): $$\frac{1-\omega^2LC}{1-\omega^2LC+j\omega LG}$$ Thanks
Vinícius Lopes Simões
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$C$ is a complex number.$f:C\to R$ is defined by $f(z)=|z^3-z+2|.$Find the maximum value of $f(z)$ if $|z|=1.$

$C$ is a complex number.$f:C\to R$ is defined by $f(z)=|z^3-z+2|.$Find the maximum value of $f(z)$ if $|z|=1.$ My try: I applied $|z^3-z+2|\leq|z|^3+|-z|+|2|$,i got $f(z)\leq 4$ but book says my answer is wrong. Correct answer is $|f(z)|$ is…
Brahmagupta
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Solving for $k$ when $\arg\left(\frac{z_1^kz_2}{2i}\right)=\pi$

Consider $$|z|=|z-3i|$$ We know that if $z=a+bi\Rightarrow b=\frac{3}{2}$ $z_1$ and $z_2$ will represent two possible values of $z$ such that $|z|=3$. We are given $\arg(z_1)=\frac{\pi}{6}$ The value of $k$ must be found assuming…
M. V
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Find the Cartesian equation of the locus described by $\arg \left(\frac{z-2}{z+5} \right)= \frac{\pi}{4}$

My working: $$ \frac{x + iy - 2}{x + iy + 5} $$ $$ \frac{(x - 2 + iy)(x+5-iy)}{(x + 5 + iy)(x+5-iy)} $$ $$ \frac{x^2+5x-ixy-2x-10+2iy+ixy+5iy+y^2}{x^2+5x-ixy+5x+25-5iy+ixy+5iy+y^2} $$ $$ \frac{x^2+3x-10+y^2+7iy}{x^2+10x+25+y^2}$$ $$ \frac…
Wooneh
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